# Measurable Function

(redirected from Lebesgue measurable function)

## measurable function

[′mezh·rə·bəl ′fəŋk·shən]
(mathematics)
A real valued function ƒ defined on a measurable space X, where for every real number a all those points x in X for which ƒ(x) ≥ a form a measurable set.
A function on a measurable space to a measurable space such that the inverse image of a measurable set is a measurable set.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Measurable Function

(in the original meaning), a function f(x) that has the property that for any t the set Et of points x, for which each f(x) ≤ t, is Lebesgue measurable. This definition of a measurable function was given by the French mathematician H. Lebesgue. The sum, difference, product, and quotient of two measurable functions, as well as the limit of a sequence of measurable functions, are in turn measurable functions. Thus, the basic operations of algebra and analysis do not go beyond the framework of the set of measurable functions. Russian and Soviet mathematicians have made a major contribution to the study of measurable functions (D. F. Egorov, N.N. Luzin, and their students). Luzin proved that a function is measurable if and only if it can be made continuous after its values are varied in a set of as small as desired measure. This is the so-called C-property of measurable functions.

In the abstract theory of measure, the function f(x) is said to be a measurable function with respect to some measure μ. if the set Et is found in the domain of definition of the measure μ. In modern probability theory measurable functions are called random variables.

References in periodicals archive ?
where f is a Lebesgue measurable function. Also, the conditional expectation of f given A, with A being a Lebesgue measurable set, is defined by E[f | A] = E[f x [[chi].sub.A]].
Suppose [alpha](t) [greater than or equal to] [[alpha].sub.0] > 0 is a Lebesgue measurable function, and at least one of the following conditions is satisfied:
A Lebesgue measurable function A is a (q, N)-atom for [H.sup.p] (R) if there exists a B 2 B such that
If 1/q - 1/r < [mu] [less than or equal to] 1/q, then for any Lebesgue measurable function a satisfying
In the general case of any Lebesgue measurable function f with bounded variation M > 0 the inequality of Lemma 3.9 is valid too, since changing its value on the set of arbitrary small measure, we get a continuous function [f.sub.1] with variation bounded by M.
Here w is a nonnegative Lebesgue measurable function defined on (0, [infinity]) and called weight.
and [PHI] is a quasi-normed space of Lebesgue measurable functions, defined on (0, [infinity]), with monotone quasi norm as follows: [absolute value of g] [less than or equal to] [absolute value of h] implies [[parallel]g[parallel].sub.[PHI]] [less than or equal to] [[parallel]h[parallel].sub.[PHI]] such that min(1, t) [member of] [PHI].
We consider vectorial measures [micro] = ([[micro].sub.0], ..., [[micro].sub.k]) in the definition of our Sobolev space and make for each one the decomposition d[micro]p; = [d([[micro].sub.j]).sub.s] + [w.sub.j]dx, where [([[micro].sub.j]).sub.s] is singular with respect to the Lebesgue measure and [w.sub.j] is a Lebesgue measurable function.
The small Lebesgue space [L.sup.(p,[alpha]] = [L.sup.(p,[alpha]]([OMEGA]) is formed by all those real-valued Lebesgue measurable functions f on [OMEGA], for which the norm
The small Lebesgue space [L.sup.(p,b,q] = [L.sup.(p,b,q]([OMEGA]) is formed by all those real-valued Lebesgue measurable functions f on [OMEGA], for which the quasi-norm
We denote by [L.sup.s](T,Z) the space of all (equivalence classes of) strongly Lebesgue measurable functions w : T [right arrow] Z such that t [right arrow] [[parallel]w(t)[parallel].sup.s] is Lebesgue integrable.
We denote by [[??].sub.M,[omega]] (T) the class of Lebesgue measurable functions f : T [right arrow] C satisfying the condition

Site: Follow: Share:
Open / Close